What do you call the nephew of your adoptive father? Is there a specific word for it? I forgot how we call them, would it be half-nephew? Because I thought half-nephew is used if your mother divorced your dad, and another dad came in and had a nephew.
Your father's nephew would be your cousin.
So your adoptive father's nephew could perhaps be referred to as your "adoptive cousin". But this man's nephew will still always be his nephew (regardless of his relationship to you), so the term "half-nephew" should not enter into the discussion at all.
For most everyday purposes, adoptive families usually function like other families, so I think in regular conversation you would normally say your adoptive father's nephew was "your cousin" and leave it at that.
Unless you are really getting technical and doing a genealogical analysis, the word "cousin" is often used loosely anyway. There are precise words for different types of cousin ("second-cousin", "cousin once removed", etc.), but most people don't really know what those terms mean. Sometimes "cousin" can refer to your cousin's cousin, or your uncle's cousin, or even the children of very close family friends, so it shouldn't be too far fetched to extend the cousin-relationship to adopted members of the family.
If it was important to distinguish between genetically related cousins and adoptive ones, you would probably use "adoptive cousin", but otherwise just plain "cousin" ought to take care of it.